If $16000 is invested at an interest rate of 6% for 25 years compounded continuously determine the annual growth rate if the money were compounded quarterly? Rounded to the nearest thousandth of a percent.
A) 0.011%
B) 6.000%
C) 6.184%
D) 0.060%
E) 106.184%

Question

Quizplus · Accepted Answer

To solve the problem, we can use the formula for continuous compounding:

$A = Pe^{rt}$

where $A$ is the final amount, $P$ is the principal, $r$ is the interest rate, and $t$ is the time in years.

For continuous compounding, we have:

$A = Pe^{0.06 \cdot 25} = 16000e^{1.5} \approx 66439.61$

Now, let's look at quarterly compounding. We can use the formula:

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

where $n$ is the number of compounding periods per year (4 in this case) and $t$ is the time in years.

We want to solve for $r$, so we can rearrange the formula to:

$r = n\left(\sqrt[n]{\frac{A}{P}}-1\right)$

Plugging in the values, we get:

$r = 4\left(\sqrt[4]{\frac{66439.61}{16000}}-1\right) \approx 0.06184$

Therefore, the annual growth rate if the money were compounded quarterly is approximately 6.184%, which is closest to choice C.

If $16000 Is Invested at an Interest Rate of 6